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Topological Vector Spaces

Maria Infusino University of Konstanz Summer Semester 2017

Contents

Introduction iii

1 Preliminaries 1

1.1 Topological spaces . . . 1

1.1.1 The notion of topological space . . . 1

1.1.2 Comparison of topologies . . . 4

1.1.3 Reminder of some simple topological concepts. . . 6

1.1.4 Mappings between topological spaces. . . 9

1.1.5 Hausdor↵spaces . . . 11

1.2 Linear mappings between vector spaces . . . 12

2 Topological Vector Spaces 15 2.1 Definition and main properties of a topological vector space . . 15

2.2 Hausdor↵topological vector spaces . . . 22

2.3 Quotient topological vector spaces . . . 24

2.4 Continuous linear mappings between t.v.s. . . 27

2.5 Completeness for t.v.s. . . 29 i

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Contents

3 Finite dimensional topological vector spaces 35

3.1 Finite dimensional Hausdor↵t.v.s. . . 35

3.2 Connection between local compactness and finite dimensionality 39 4 Locally convex topological vector spaces 41 4.1 Definition by neighbourhoods . . . 41

4.2 Connection to seminorms . . . 46

4.3 Hausdor↵locally convex t.v.s . . . 56

4.4 The finest locally convex topology . . . 59

4.5 Finite topology on a countable dimensional t.v.s. . . 61

4.6 Continuity of linear mappings on locally convex spaces . . . 63

5 The Hahn-Banach Theorem and its applications 65 5.1 The Hahn-Banach Theorem . . . 65

5.2 Applications of Hahn-Banach theorem . . . 69

5.2.1 Separation of convex subsets of a real t.v.s. . . 70

5.2.2 Multivariate real moment problem . . . 72

Bibliography 77

ii

Referenzen

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